7 THE STURM-LIOUVILLE PROBLEM 353

(b) Necessity. Apply the identity (11,7.5) in both intervals a < t ^ x
and x^t^b, with u(t) = y(t) and v(t) = Kx(t); the relation y(x) =
- pKO, .*)/(/) A follows at once from the properties (11.7.6) of the Green
function.

From (11.7.8) it follows that any solution of the Sturm-Liouville problem
is a solution of the Fredholm integral equation with hermitian kernel:

(1 1 .7.9) y(x) - J, K(/, x)y(t) A =

where

and conversely. The inverses Xn of the eigenvalues ^ 0 of the operator U
in the prehilbert space G defined in (11.2.8), corresponding to the kernel
function K, are called the eigenvalues of the Sturm-Liouville problem. We
can now state the following theorem which solves the Sturm-Liouville prob-
lem in every case :

(1 1 .7.1 0) For any real-valued continuous function q(x) in the compact interval
I = [a,b]:

(a) The Sturm-Liouville problem has an infinite strictly increasing sequence
of eigenvalues (A,,) which are real numbers such that lim An = •+ oo and that the

fl~»-OQ

series Y, 1/AjJ is convergent.

n

(b) For each eigenvalue kn , the homogeneous Sturm-Liouville problem has
a real valued solution cpn(x) such that <p%(x) dx = 1 , and every other solution

is a constant multiple of(pn .

(c) The sequence ((pn) is a total orthonormal system in the prehilbert space
G (notation of Section 11.6).

(d) Let w be a complex-valued continuous function in I, which is the
primitive of a regulated function w', such that: (i) wf is continuous in I except at
a finite number of interior points; (ii) wf has a continuous derivative w" in each
interval in which it is continuous; (iii) w satisfies the boundary conditions

(11.7.2). Then, if cn = (w\ <pn) = f w(t)<pn(t) dt, we have vv(x) = £ c,, <?„(*)

Ja n

where the series is absolutely and uniformly convergent in I.

(e) If A is not one of the eigenvalues Xn ,for each regulated function f, con-
tinuous in I except at a finite number of points, the Sturm-Liouville problem-
has a unique solution w which is such that cn = (w\ cpn) is given by the formula

cn = dn/(l - AJ, where dn = /«<?„« dt.(b) Kt satisfies the boundary conditions (11.7.2).